tell me about mises transfomation? Why it is need? and what merrit it has?

There can be more than one Mises transformation. I know one only:

The boundary layer equations involve two unknowns, u and v, and two independent variables, x and y (y is the distance to the wall). Correspondingly, there are two equations: momentum and continuity. If, instead of x and y, you use x and \psi (\psi is the stream function) then it turns out that v drops out of the momentum equation (using continuity equation, of course, to do that). This is the Mises transformation. The advantage is that instead of two equations you have only one, the momentum equation for u(x,\psi). Also, in certain cases the boundary layer thickness varies less in Mises variables. The disadvantage is that u(\psi) is singular at the wall. A workaround is to approximate the second derivative in y variables but using the grid in \psi variable. Another disadvatage is that Mises variables are difficult to use for the triple-deck theory equations with reversed flow present.

Sergei